Open book decompositions of links of quotient surface singularities and support genus problem

TL;DR

This paper explicitly constructs open book decompositions for links of quotient surface singularities, demonstrating minimal page-genus and exploring the relationship between Milnor genus and support genus.

Contribution

It provides explicit open book decompositions supporting Milnor fillable contact structures and investigates the equality of Milnor and support genus for these links.

Findings

Milnor open books have minimal page-genus among all supporting the same contact structure.

For many quotient surface singularities, Milnor genus equals support genus.

An upper bound for support genus is found in remaining cases.

Abstract

In this paper we write explicitly the open book decompositions of links of quotient surface singularities supporting the corresponding unique Milnor fillable contact structure. The page-genus of these Milnor open books are minimal among all Milnor open books supporting the same contact structure. We also investigate whether the Milnor genus is equal to the support genus for links of quotient surface singularities. We show that for many types of the quotient surface singularities the Milnor genus is equal to the support genus. In the remaining cases we are able to find a small upper bound for the support genus.